Finite Group Databasebeta

← Prev (#839) Next (#841) →

C₃×#97 #840

order	96
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#2021 (order 256) · Inn(G) = C₂² · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₂²
commutator	C₂	C₃×C₄²
Frattini, cosocle	C₂×C₄	C₂²×C₃
socle, prime radical	C₂²×C₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 2 generators on 15 points:

(5,6,7,10,8,9,11,12)(13,14,15);(1,2,3,4)(6,9)(10,12)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	C₉×#1355, C₉×#1356, #11771983, …
Fitting subgroup of	#4464, #4465, #4466, …
maximal subgroup of	#4464, #4465, #4466, …
maximal normal subgroup of	#4464, #4465, #4466, …
maximal quotient of	C₃×#322, C₃×#336, C₃×#347, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 47);